Although she has made some terrible moves, such as passing a law that denies drivers' licenses and other public benefits to undocumented immigrants, she at least showed that she has, somewhere deep in her, a sense of justice. In essence, she realized that signing SB 1062 would have given a lot of people lots of power to discriminate in all sorts of ways. For example, a Muslim taxi driver in Tucson could refuse to pick up a woman who's traveling alone.
Plus, I think she's a sensible enough person to realize that, well, someone whom you think is gay or Muslim or whatever may, in fact, not be. I know personally people about whom you would "never guess" their sexual orientation or religion, or even race or nationality. (Not many people think, in looking at me, that most of my heritage is Italian---Sicilian at that! And I know a black man who looks just about as white as I do.) Any business owner who discriminates on the basis of mistaken identity is practically setting him or her self up for a lawsuit.
Furthermore, she surely realized that signing SB 1062 would be bad for business in The Grand Canyon State. The Hispanic National Bar Association had already announced that it cancelled plans to hold its annual convention--which around 2000 would have attended--in the state next year, in part as a response to the bill. And the National Football League--not exactly known as a bastion of gay-rights advocacy--said it was exploring plans to move the Super Bowl, which is to be held in the state next year, to a different venue had Brewer signed the bill into law.
Finally, I think she may have had a personal motivation (i.e., guilt) for signing SB 1062. In 2007, one of her sons died of cancer and HIV-related illnesses. Various accounts say that she disowned him when he "came out" and, at the time of his death, hadn't spoken to him in years. Perhaps she is doing in his memory what she didn't do for him during his life.